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📰 Question: An environmental consultant models a river's flow as the line $y = -\frac{1}{2}x + 5$. Find the point on this line closest to the pollution source at $(4, 3)$. 📰 Solution: The closest point is the projection of $(4, 3)$ onto the line. The formula for the projection of a point $(x_0, y_0)$ onto $ax + by + c = 0$ is used. Rewriting the line as $\frac{1}{2}x + y - 5 = 0$, we compute the projection. Alternatively, parametrize the line and minimize distance. Let $x = t$, then $y = -\frac{1}{2}t + 5$. The squared distance to $(4, 3)$ is $(t - 4)^2 + \left(-\frac{1}{2}t + 5 - 3\right)^2 = (t - 4)^2 + \left(-\frac{1}{2}t + 2\right)^2$. Expanding: $t^2 - 8t + 16 + \frac{1}{4}t^2 - 2t + 4 = \frac{5}{4}t^2 - 10t + 20$. Taking derivative and setting to zero: $\frac{5}{2}t - 10 = 0 \Rightarrow t = 4$. Substituting back, $y = -\frac{1}{2}(4) + 5 = 3$. Thus, the closest point is $(4, 3)$, which lies on the line. $\boxed{(4, 3)}$ 📰 Question: A hydrologist models groundwater flow with vectors $\mathbf{a} = \begin{pmatrix} 2 \\ -3 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ 4 \end{pmatrix}$. Find the angle between these flow directions. 📰 Number Of Ways To Choose 1 Simulation Technique 2888658 📰 Steven Cree Movies And Tv Shows 8912997 📰 How Many Years Before You Can Cash Out Your 401K The Shocking Answer Will Change Everything 5048722 📰 Beatking 3666245 📰 Crazygames Com Bloxdio 5 Impossible Tricks That Will Blow Your Mind 9789093 📰 What Happens If I Only Get The Powerball Number 4867473 📰 This Map Unlocks Secrets Behind Every Region In Genshin Interactiveyoull Be Astounded 888254 📰 Breaking Weatherunder Analysis Shocks Every Trendsetter This Season 2365396 📰 Ravioli Vs Myth The Truth About Filling And Flavor You Demand 9811415 📰 Yeastie Boys 6223051 📰 No Te Imaginas Lo Que Descubrir Al Entender El Espaol De Verdad 8298318 📰 Swivel Made Easy The Sleekest Leather Chair That Quietly Dominates Every Room 143465 📰 Apple Student Discount 1660531 📰 Shocking Wood Block Puzzle Online Solve It Before You Miss The Fun 4481190 📰 Tuscola County 2624337